Two Proofs of Fine's Theorem

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Fine's theorem concerns the question of determining the conditions under which a certain set of probabilities for pairs of four bivalent quantities may be taken to be the marginals of an underlying probability distribution. The eight CHSH inequalities are well-known to be necessary conditions, but Fine's theorem is the striking result that they are also a sufficient condition. It has application to the question of finding a local hidden variables theory for measurements of pairs of spins for a system in an EPRB state. Here we present two simple and self-contained proofs of Fine's theorem in which the origins of this non-obvious result can be easily seen. The first is a physically motivated proof which simply notes that this matching problem is solved using a local hidden variables model given by Peres. The second is a straightforward algebraic proof which uses a representation of the probabilities in terms of correlation functions and takes advantage of certain simplifications naturally arising in that representation. A third, unsuccessful attempt at a proof, involving the maximum entropy technique is also briefly described